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Question 249138: Problem goes as follows:
"Graph the line x=4"
the answer provided is as follows:
(4,-5)(4,-1)(4,7).
I do not understand how this answer was discovered. I am in dire need of your assistance. Any help is immensely appreciated! Thank you.
Found 2 solutions by jim_thompson5910, Theo:
Answer by jim_thompson5910(35256)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! x = 4 is a vertical line.
the slope is undefined because there is never a change in the value of x for a change in the value of y.
the slope of the equation is (y2-y1) / (x2-x1)
x2 and x1 are the same so the denominator of the equation is 0 which leads to an undefined slope because division by 0 causes the result to be infinity which is an undefined number.
the standard form of the equation for a straight line is:
ax + by = c, where:
a = the coefficient of the x term
b = the coefficient of the y term
c = the constant.
you can convert the standard form of the equation to the slope intercept form by doing the following:
ax + by = c
subtract ax from both sides to get:
by = -ax + c
divide both sides by b to get:
y = (-a/b)*x + (c/b)
that's the slope intercept form of the equation of a straight line.
(-a/b) = the slope of the line, called m.
(c/b) = the y-intercept of the line, called b
the slope intercept form of the equation is called y = m*x + b, where:
m is the slope
b is the y-intercept.
In the standard form of the equation or:
ax + by = c,
if b = 0, then the equation becomes:
ax = c, because the y terms drops out (anything multiplied by 0 equals 0).
Because the y term drops out of the equation, the value of y is not constrained to be anything special in relationship to x.
It is not limited by the rules of the equation.
this means that y can be anything, but x has to be what the equation says it has to be.
when you graph the equation, you are plotting the value of x in relation to the value of y.
x = 4 means that the value of x has to be equal to 4.
y not in the equation means that y is free to be any value it wants.
this is what creates the vertical line.
when you plot the equation of a vertical line, you can see that the value of x has to be 4.
you can also see that, along that line, the value of y can be any value.
that is why your solution showed you (4,a), (4,b), (4,c), etc.
there was no equation to force the selection of y given a specific value of x.
for example, if the equation was y = 4x, then when x = 1, y has to be 4*1 and could be nothing else.
In the example x = 4, there is no such restriction for the value of y, meaning the value of y could be anything, but the value of x had to be 4.
when you plot the line you will see that it is a vertical line on the graph.
the requirements of the equation are that x = 4.
there are no requirements on the value of y, so pick any value of y when x = 4 and create your vertical line.
if your equation was y = 5, then you would have a horizontal line.
this might be easier for you to see.
In that case, x can be any value and y always has to be 5, regardless of the value of x.
x = 4 is the same concept. y can be any value and x always has to be 4, regardless of the value of y.
the only difference is, that in the case of y = 5, the slope is equal to 0, while in the case of x = 4, the slope is undefined (division by 0 caused undefined).
look at the standard form of the equation again:
ax + by = c.
when b = 0, y drops out of the equation, and when a = 0, x dropes out of the equation.
you get:
ax = c when y drops out.
you get:
by = c when x drops out.
since the slope is (-a/b), when b = 0, the slope is undefined because you are dividing by 0, and when a = 0, the slope is 0 because you are dividing into 0.
best answer I can give. i don't know how to explain it any other way.
the answer is:
since you are plotting a graph of y in relationship to x, since y is not in the equation, y can be any value as long as x = 4 because the only constraint in the equation is that x = 4. y, not being in the equation, has no constraints. you are free to pick any value of y as long as x = 4.
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